Positive representations, multiplier Hopf algebra, and continuous canonical basis (String theory, integrable systems and representation theory)

"String theory, integrable systems and representation theory". July 30～August 2, 2013. edited by Koji Hasegawa and Yasuhiko Yamada. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.We introduce the language of multiplier Hopf algebra in the context of positive representations of split real quantum groups, and discuss its applications with a continuous version of Lusztig-Kashiwara's canonical basis, which may provide a key to prove the closure of the positive representations under tensor products, and harmonic analysis of quantized algebra of functions in the sense of locally compact quantum groups